Proof of Theorem cdlemg10a
Step | Hyp | Ref
| Expression |
1 | | simp11 1204 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | | simp12 1205 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
3 | | simp13 1206 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
4 | | simp21 1207 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐹 ∈ 𝑇) |
5 | | simp22 1208 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐺 ∈ 𝑇) |
6 | | simp23 1209 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≠ 𝑄) |
7 | | simp31 1210 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄)) |
8 | | cdlemg8.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
9 | | cdlemg8.j |
. . . 4
⊢ ∨ =
(join‘𝐾) |
10 | | cdlemg8.m |
. . . 4
⊢ ∧ =
(meet‘𝐾) |
11 | | cdlemg8.a |
. . . 4
⊢ 𝐴 = (Atoms‘𝐾) |
12 | | cdlemg8.h |
. . . 4
⊢ 𝐻 = (LHyp‘𝐾) |
13 | | cdlemg8.t |
. . . 4
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
14 | 8, 9, 10, 11, 12, 13 | cdlemg9 38303 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ ((((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))) ∨ (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄)))) |
15 | 1, 2, 3, 4, 5, 6, 7, 14 | syl133anc 1394 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ ((((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))) ∨ (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄)))) |
16 | 8, 11, 12, 13 | ltrnel 37808 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊)) |
17 | 1, 5, 2, 16 | syl3anc 1372 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊)) |
18 | 8, 11, 12, 13 | ltrnel 37808 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊)) |
19 | 1, 5, 3, 18 | syl3anc 1372 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊)) |
20 | | simp12l 1287 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴) |
21 | | simp13l 1289 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴) |
22 | 11, 12, 13 | ltrn11at 37816 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → (𝐺‘𝑃) ≠ (𝐺‘𝑄)) |
23 | 1, 5, 20, 21, 6, 22 | syl113anc 1383 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ≠ (𝐺‘𝑄)) |
24 | | simp32 1211 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄)) |
25 | | cdlemg10.r |
. . . . . . . 8
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
26 | 8, 9, 10, 11, 12, 13, 25 | cdlemg10c 38308 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑅‘𝐹) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) ↔ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) |
27 | 1, 2, 3, 4, 5, 26 | syl122anc 1380 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑅‘𝐹) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) ↔ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) |
28 | 24, 27 | mtbird 328 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ¬ (𝑅‘𝐹) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) |
29 | 8, 9, 10, 11, 12, 13, 25 | trlval4 37857 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊) ∧ ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊)) ∧ ((𝐺‘𝑃) ≠ (𝐺‘𝑄) ∧ ¬ (𝑅‘𝐹) ≤ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))) → (𝑅‘𝐹) = (((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) ∧ ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄))))) |
30 | 1, 4, 17, 19, 23, 28, 29 | syl132anc 1389 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐹) = (((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) ∧ ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄))))) |
31 | | simp11l 1285 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL) |
32 | 8, 11, 12, 13 | ltrnat 37809 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴) |
33 | 1, 5, 20, 32 | syl3anc 1372 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ∈ 𝐴) |
34 | 8, 11, 12, 13 | ltrnat 37809 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) |
35 | 1, 4, 33, 34 | syl3anc 1372 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) |
36 | 9, 11 | hlatjcom 37037 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝐺‘𝑃) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) → ((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) = ((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃))) |
37 | 31, 33, 35, 36 | syl3anc 1372 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) = ((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃))) |
38 | 8, 11, 12, 13 | ltrnat 37809 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐺‘𝑄) ∈ 𝐴) |
39 | 1, 5, 21, 38 | syl3anc 1372 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑄) ∈ 𝐴) |
40 | 8, 11, 12, 13 | ltrnat 37809 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝐺‘𝑄) ∈ 𝐴) → (𝐹‘(𝐺‘𝑄)) ∈ 𝐴) |
41 | 1, 4, 39, 40 | syl3anc 1372 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐹‘(𝐺‘𝑄)) ∈ 𝐴) |
42 | 9, 11 | hlatjcom 37037 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝐺‘𝑄) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑄)) ∈ 𝐴) → ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄))) = ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))) |
43 | 31, 39, 41, 42 | syl3anc 1372 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄))) = ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))) |
44 | 37, 43 | oveq12d 7200 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))) ∧ ((𝐺‘𝑄) ∨ (𝐹‘(𝐺‘𝑄)))) = (((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄)))) |
45 | 30, 44 | eqtrd 2774 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐹) = (((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄)))) |
46 | | simp33 1212 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄)) |
47 | 8, 9, 10, 11, 12, 13, 25 | trlval4 37857 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ (𝑄 ∨ (𝐺‘𝑄)))) |
48 | 1, 5, 2, 3, 6, 46,
47 | syl132anc 1389 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ (𝑄 ∨ (𝐺‘𝑄)))) |
49 | 9, 11 | hlatjcom 37037 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐺‘𝑃)) = ((𝐺‘𝑃) ∨ 𝑃)) |
50 | 31, 20, 33, 49 | syl3anc 1372 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ (𝐺‘𝑃)) = ((𝐺‘𝑃) ∨ 𝑃)) |
51 | 9, 11 | hlatjcom 37037 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐺‘𝑄) ∈ 𝐴) → (𝑄 ∨ (𝐺‘𝑄)) = ((𝐺‘𝑄) ∨ 𝑄)) |
52 | 31, 21, 39, 51 | syl3anc 1372 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑄 ∨ (𝐺‘𝑄)) = ((𝐺‘𝑄) ∨ 𝑄)) |
53 | 50, 52 | oveq12d 7200 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ (𝑄 ∨ (𝐺‘𝑄))) = (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄))) |
54 | 48, 53 | eqtrd 2774 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) = (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄))) |
55 | 45, 54 | oveq12d 7200 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) = ((((𝐹‘(𝐺‘𝑃)) ∨ (𝐺‘𝑃)) ∧ ((𝐹‘(𝐺‘𝑄)) ∨ (𝐺‘𝑄))) ∨ (((𝐺‘𝑃) ∨ 𝑃) ∧ ((𝐺‘𝑄) ∨ 𝑄)))) |
56 | 15, 55 | breqtrrd 5068 |
1
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ ((𝑅‘𝐹) ∨ (𝑅‘𝐺))) |